Speeding the Pollard and elliptic curve methods of factorization

Elliptic Curve Method (ECM) is a well-known method for factoring integers, which is usually used in the Number Field Sieve algorithm as a subroutine for factoring smaller integers than the targeted one. ECM is called many times and can be executed in parallel for different inputs. This method mainly consist of simple operations on elliptic curves. Thus, ECM is suitable for hardware implementations that can efficiently reduce computational time. This work describes a new, improved FPGA-based hardware accelerator for ECM, designed for large scale computations. Our accelerator can operate with an on board ARM processor or with an external host computer. This design can factor several numbers at once and can be easily ported to various FPGA boards. Different methods for improving results (e.g. the use of DSP blocks, cache-registers, reorganizing instruction order) are described and their performance is analyzed. As a result, one of the fastest hardware ECM units is achieved.

Elliptic curve cryptosystem is used in the process of embedded systems, the selection and generation algorithm of the elliptic curve will directly affect the efficiency of systems. From Elliptic Curve's selection, Elliptic Curve's structure, Elliptic Curve's generation, this paper discussed the realization of a random elliptic curve method of Embedded Mobile Device, the SEA algorithm and its improved algorithm. The results show that this method can achieve a quick implementation of the elliptic curve method to improve the operating efficiency of embedded systems in the same security guarantees.

Several mathematical concepts from analytics number theory, especially the counting functions (CFs) are employed to solve some modern problems in mathematics, computer sciences, engineering and others. One of them is the factorization problem for integers into two primes. This work proposes an extended factorization problem using the CFs, as main part from analytics number theory, to factor a real number r into two primes p and q. The computations of the points doubling and addition on the elliptic curves E modulo a composite number n are also used in this extension to give new two versions of the factorization over real field R. On the first version, the primes are obtained from doubling and addition on elliptic curve points that are computed modulo composite number which is resulting as output of the CFs algorithm with the input r. Whereas, the primes, on the second version, are got directly from applying the CFs algorithm with r as input. The successful computation of the scalar multiplication kP on the elliptic curve E modulo primes are determined. The computational complexities of extended elliptic curve factorization versions are determined using the counting operations. These operations are elliptic curve and finite field operations. New numerical results on two versions are computed as study cases. The elliptic curve factorization algorithm over R (ECF-R) with two versions, is more efficient for factoring-based cryptography in comparing to the original ECM.

The Enhanced and Secure RSA Key Generation Scheme (ESRKGS), introduced in 2014, aimed to improve RSA security by employing a modulus constructed from four prime factors. However, subsequent studies in 2016 revealed that this structure did not provide additional security over standard RSA. In response, a modified version of ESRKGS was proposed in 2021, incorporating dual encoding techniques using a masking parameter γ and double encryption. This study evaluates the security of the modified ESRKGS by simulating an attack scenario in which the adversary is assumed to know of ϕ(N ), enabling recovery of encrypted messages. Additionally, we implement Lenstra’s Elliptic Curve Method (ECM) to assess the factorization resistance of the four-prime modulus when ϕ(N ) is not known. Experimental results indicate that ECM can efficiently factor the modulus into its four constituent primes under practical time constraints. These findings demonstrate that, despite recent modifications, the ESRKGS variant remains vulnerable to factorization based attacks. This highlights the necessity for more rigorous cryptographic design principles in multiprime RSA systems and calls into question the long-term viability of ESRKGS-based schemes in high-security applications.

Various cryptosystems have been proposed whose security relies on the difficulty of factoring integers of the special form N = pq 2. To factor integers of that form, Peralta and Okamoto introduced a variation of Lenstra’s Elliptic Curve Method (ECM) of factorization, which is based on the fact that the Jacobi symbols (a/N) and (a/P) agree for all integers a coprime with q. We report on an implementation and extensive experiments with that variation, which have been conducted in order to determine the speed-up compared with ECM for numbers of general form.KeywordsPrime FactorElliptic CurveEquality CheckGreat Common DivisorDecimal DigitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We describe the complete factorization of the tenth Fermat number F 10 F_{10} by the elliptic curve method (ECM). F 10 F_{10} is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 , … , F 11 F_5, \dots , F_{11} .

We describe an algorithm for proving that there is no odd perfect number less than a given bound K (or finding such a number if one exists). A program implementing the algorithm has been run successfully with K = 10 160 K = {10^{160}} , with an elliptic curve method used for the vast number of factorizations required.

We report the discovery of a new factor for each of the Fermat numbers F 13 , F 15 , F 16 F_{13}, F_{15}, F_{16} . These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with 2391 2391 , 9808 9808 and 19694 19694 decimal digits respectively.

The problem of factorization of large integers into the prime factors has always been of mathematical interest for centuries. In this paper, starting with a historical overview of integer factorization algorithms, the study is extended to some recent developments in the prime factorization with Pisano period. To reduce the computational complexity of Fibonacci number modulo operation, the fast Fibonacci modulo algorithm has been used. To find the Pisano periods of large integers, a stochastic algorithm is adopted. The Pisano period factorization method has been proved slightly better than the recently developed algorithms such as quadratic sieve method and the elliptic curve method. This paper ideates new insights in the area of integer factorization problems.KeywordsCryptographyNetwork securityInteger factorization algorithmPisano period

In this paper a probabilistic algorithm is exhibited that factors any positive integer n n into prime factors in expected time at most L n [ 1 2 , 1 + o ( 1 ) ] {L_n}[\tfrac {1}{2},1 + o(1)] for n → ∞ n \to \infty , where L x [ a , b ] = exp ( b ( log ⁡ x ) a ( log log ⁡ x ) 1 − a ) {L_x}[a,b] = {\text {exp}}(b{(\log x)^a}{({\text {log}}\log x)^{1 - a}}) . Many practical factoring algorithms, including the quadratic sieve and the elliptic curve method, are conjectured to have an expected running time that satisfies the same bound, but this is the first algorithm for which the bound can be rigorously proved. Nevertheless, this does not close the gap between rigorously established time bounds and merely conjectural ones for factoring algorithms. This is due to the advent of a new factoring algorithm, the number field sieve, which is conjectured to factor any positive integer n n in time L n [ 1 3 , O ( 1 ) ] {L_n}[\tfrac {1}{3},O(1)] . The algorithm analyzed in this paper is a variant of the class group relations method, which makes use of class groups of binary quadratic forms of negative discriminant. This algorithm was first suggested by Seysen, and later improved by A. K. Lenstra, who showed that the algorithm runs in expected time at most L n [ 1 2 , 1 + o ( 1 ) ] {L_n}[\tfrac {1}{2},1 + o(1)] if one assumes the generalized Riemann hypothesis. The main device for removing the use of the generalized Riemann hypothesis from the proof is the use of multipliers. In addition a character sum estimate for algebraic number fields is used, with an explicit dependence on possible exceptional zeros of the corresponding L L -functions. Another factoring algorithm using class groups that has been proposed is the random class groups method. It is shown that there is a fairly large set of numbers that this algorithm cannot be expected to factor as efficiently as had previously been thought.

An addition law for an elliptic curve is complete if it is defined for all possible pairs of input points on the elliptic curve. In Elliptic Curve Cryptography (ECC), a complete addition law provides a natural protection against side-channel attacks which are based on Simple Power Analysis (SPA). Montgomery curves are a specific family of elliptic curves that play a crucial role in ECC because of its well-known Montgomery ladder, particularly in the Elliptic Curve Diffie-Hellman Key Exchange (ECDHKE) protocol and the Elliptic Curve factorization Method (ECM). However, the complete addition law for Montgomery curves, as stated in the literature, has a computational cost of 14M+ 2D, where M,D denote the costs of a field multiplication and a field multiplication by a constant, respectively. The lack of a competitive complete addition law has led implementers towards twisted Edwards curves, which offer a complete addition law at a lower cost of 8M+ 1D for appropriately chosen curve constants.In this paper, we introduce extended Montgomery coordinates as a novel representation for points on Montgomery curves. This coordinate system enables us to define birational multiplication-free maps between the extended twisted Edwards coordinates and extended Montgomery coordinates. Using this map, we can transfer the complete addition laws from twisted Edwards curves to Montgomery curves without incurring additional multiplications or squarings. In addition, we employ a technique known as scaling to refine the addition laws for twisted Edwards curves, which results in having i) Complete addition laws with the costs varying between 8M+1D and 9M+1D for a broader range of twisted Edwards curves, ii) Incomplete addition laws for twisted Edwards curves with the cost of 8M. Consequently, by leveraging our birational multiplication-free maps, we present complete addition laws for Montgomery curves with the cost of 8M+1D. This shows a significant improvement for complete addition law for Montgomery curves by reducing the computational cost by 6M+ 1D. This improvement makes Montgomery curves a more attractive option for applications where an efficient complete addition law is essential.

Fast and rigorous factorization under the generalized Riemann hypothesis

An anonymity-preserving mobile user authentication protocol for global roaming services

The performance of seven different methods (Differential, Fujimoto, Thomas, Graphical, Integral, Log-Difference, and Nonlinear Regression) for estimating first-stage, carbonaceous biochemical oxygen demand (CBOD), curve parameters, namely k and L0, were compared using synthetic data generated by Monte Carlo simulation technique. The comparison of the methods was made based on their efficiency in retrieving the original values of k and L0, which were selected to generate the synthetic data. In the first part of the study, five sets of “true” data (without error substitution) with different k and L0 value pairs, (k (d−1)-L0 (mg l−1): 0.23-10,000; 0.23-250; 0.23-50; 0.10-250; and 0.50-250) were used to obtain information about the effect of different k-L0 combinations and of using 5-day and 20-day CBOD data on the performance of the methods. In the second part, the same methods were used to calculate k and L0 for ten sets of synthetic data with log-normally distributed random errors at the coefficient of variation (COV) levels of 0.1, 0.2, and 0.3 for a single k-L0 value pair, (0.23 d−1; 250 mg l−1). The results indicated that: (1) different combinations of k-L0 values had no significant effect on the performance of CBOD curve parameter estimation methods with the “true” data; (2) use of CBOD20 data, i.e., CBOD data collected for 20 days, provided better estimates for k and L0; (3) the Integral and Nonlinear Regression techniques were found to be the most reliable methods for the estimation of CBOD curve parameters among the other methods considered in this study.

Eichler integrals play an integral part in the modular parametrizations of elliptic curves. In her master’s thesis, Kodgis conjectures several dozen zeros of Eichler integrals for elliptic curves with conductor ≤ 179. In this paper we prove a general theorem which confirms many of these conjectured zeros. We also provide two ways to generate infinite families of elliptic curves with certain zeros of their Eichler integrals.